Tests for Linearity

For each experiment, the left and right sides of Equation 5-323 are determined. The values of botli sides are plotted and tested for linearity. 

In our next chapter we will discuss other methods of testing for linearity that have appeared in the literature. Afterward, we will then turn our attention to a new test that has been devised. In fact, it turns out that while DW has much to recommend it, it is not the final or best answer. The new method, however, is much more direct and specific even than DW. It is the correct way to test for linearity. We will discuss it all in due course, in this same place. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

The method development process with the multisorbent plate consists of three steps. In step 1, the sorbent chemistry and the pH for loading, washing, and elution are optimized. In step 2, optimization of the percentage organic for wash and elution and the pH of the buffer needed is carried out. Step 3 is validation the method developed from the results of the previous two steps is tested for linearity, limits of detection, quantitation of recovery, and matrix effects using a stable isotope-labeled analyte as an IS. 

The relationship between doses of a chemical and measured parameters, usually including tests for linearity. 

Armitage, P. (1955). Tests for linear trends in proportions and frequencies. Biometrics. 11 375-386. 

The amount transformation process is illustrated with data for chlorpyrifos in the flame photometric detector, phosphorus mode, and shown in Table VI. Level 1 transformations were calculated where the amount power was increased by 0.03 units for each step. At an amount power of 0.20 the F statistic of 32.7 showed a minimum but at a confidence level of 95% did not satisfy the F test for linearity. Power steps changed by only 0.01 and 0.001 units in the vicinity of the minimum were then calculated as shown in levels 2 and 3. The best linearity was found in this case at a power transformation of 0.182 although the F statistic of 8.33 did not indicate linearity when compared with the critical F of 2.99 at P=.95. Calculations at these second and third levels were not always necessary and even when performed did not always lead to a satisfactory condition of linearity. 

Table VII displays the results of the test for linearity after transforming the amount to a minimum F statistic.

In a recent study, Mirza et al. have pointed out that despite the obvious advantages, fiber-based dissolution studies are not yet implemented in industry. They have validated a fiber-optic based dissolution test for linearity, precision, accuracy, specificity and robusmess, with excellent results. 

Other Statistical Tests for Linearity Several other tests are for linearity have been described in the literature the test for significance of the quadratic... 

In order to minimize external (bed) diffusion resistance and maximize the heat transfer rate it is desirable to use a very small adsorbent sample with the crystals spread as thinly as possible over the balance pan or within the containing vessel. To minimize the effect of non-linearities, such as the strong concentration dependence of the diffusivity, measurements should be made differentially over small concentration changes. Variation of the step size and comparison of adsorption and desorption curves provide simple tests for linearity of the system. The large differences between adsorption and desorption diffusivities, reported in some of the earlier work, have been shown to be due to the concentration dependence of the diffusivity(8) and in differential measurements under similar conditions no such anomaly was observed. 

Test for linearity and repeatability (test sequence of Section 25.3). The required number of vials are put into the autosampler, the test sequence is programmed and the test is started. It is absolutely necessary to work with both adequate time constant of the detector and integration parameters. 

If TLC is used as an analytical method in quality control, the reproducibUity, i.e. lack of scatter (precision of both system and method), and also the accuracy of the analytical results must be determined. The GMP/GLP guidelines also require vahdation of the method, i.e. testing for linearity, selectivity, robustness and limits of detection and determination (see also Section 9.1 Validation of TLC Methods ). For method development, high demands are placed on the stationary phases of the chromatographic system ... 

For reversible reactions the problem of interpretation becomes more difficult since we have basically the same number of data, conversion versus time, but two rate constants to determine. If eonversion data are available up to the point of equilibrium, we may use the equilibrium eomposition together with conversion-time data to solve for the two constants directly. Generally, however, we must assume that the experimenter does not have the time or patience to obtain true equilibrium information (bearing in mind that rates of reaetion are very slow at this point) so that a simple trial procedure is probably most eonvenient. For the first-order forward and reverse case of equation (1-61) one would choose a value of a, which is a function of both kf and k, and test for linearity by plotting In [a/ a — x)] versus time. The correct choice of a will result in a straight line of slope kf + k ), and then the values for a and the slope may be used to solve for the individual constants. If equilibrium data are available, the rate of reaction is zero and the rate equation used directly to obtain the equilibrium constant ... 

After the testing for linearity (at typically 0.2 or 1 mg/mL of microsomal proteins and a 15- or 30-min period of incubation), experiments to determine other parameters can be designed. The rationale for a suitable assay condition is that such a condition will permit the formation of quantifiable amounts of the metabolites without markedly depleting (i.e., by less than 20%) the substrate. 

Empirical relationships that appear to be linear can be tested for linearity by a number of approaches. Visual review of a plot is most appropriate to reveal any departures from linearity. A plot of residuals (residual = observed - computed response value) with respect to the level of the response should not show any correlation or systematic behavior. Such a review requires at least 7 pairs of data points (response levels) to be useful. A simple F test may akso be employed. If Sf, is the pooled variance for a set of repetitive instrument responses, each set of responses at one of a series of levels (true values) of the calibration standard, and is the variance of points about the fitted function, when individual set response values (not means or averages) arc used for the least-squares calculation, then the variance ratio... 

In the majority of the literature tests for linear programming, bound constraints are rarely provided and often only nonnegativity bounds are given (this is required by the Simplex and Interior Points methods). 

The samples were cycled between 732° C and 1010°C for two weeks, for a total of 125 cycles. After cycling, the furnace temperature was raised to 1177°C to simulate volatilization of Na2S04 during a checker burnout. The samples were tested for % linear change, % weight change, and modulus of rupture at 1038° C. 

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